Universal quadratic form explained

In mathematics, a universal quadratic form is a quadratic form over a ring that represents every element of the ring.^[1] A non-singular form over a field which represents zero non-trivially is universal.^[2]

Examples

• Over the real numbers, the form x² in one variable is not universal, as it cannot represent negative numbers: the two-variable form over R is universal.
• Lagrange's four-square theorem states that every positive integer is the sum of four squares. Hence the form over Z is universal.
• Over a finite field, any non-singular quadratic form of dimension 2 or more is universal.^[3]

Forms over the rational numbers

The Hasse–Minkowski theorem implies that a form is universal over Q if and only if it is universal over Q_p for all p (where we include, letting Q_∞ denote R).^[4] A form over R is universal if and only if it is not definite; a form over Q_p is universal if it has dimension at least 4.^[5] One can conclude that all indefinite forms of dimension at least 4 over Q are universal.^[4]

See also

• The 15 and 290 theorems give conditions for a quadratic form to represent all positive integers.

References

• Book: Lam, Tsit-Yuen . Tsit Yuen Lam

. Introduction to Quadratic Forms over Fields . 67 . . Tsit Yuen Lam . American Mathematical Society . 2005 . 0-8218-1095-2 . 1068.11023 . 2104929 .

• Book: Rajwade, A. R. . Squares . 171 . London Mathematical Society Lecture Note Series . . 1993 . 0-521-42668-5 . 0785.11022 .
• Book: Serre, Jean-Pierre . Jean-Pierre Serre

. Jean-Pierre Serre . A Course in Arithmetic . . 7 . . 1973 . 0-387-90040-3 . 0256.12001 . registration .

Notes and References

1. Lam (2005) p.10
2. Rajwade (1993) p.146
3. Lam (2005) p.36
4. Serre (1973) p.43
5. Serre (1973) p.37